Per Alexandersson: Rational Schur functions and stretched Kostka coefficients

Abstract: We introduce a linear map on symmetric functions that "divides" a partition by a positive integer k, sending a Schur function indexed by a partition of kn to a symmetric function indexed by partitions of n. We show that this image is always Schur positive, meaning it expands with non-negative integer coefficients in the Schur basis. The coefficients are counted by a new family of combinatorial objects: k-Yamanouchi
tableaux, which generalize the classical ballot (Yamanouchi) tableaux appearing in the Littlewood–Richardson rule.

A second, more surprising application yields a connection to work of Amdeberhan–Shareshian–Stanley on alternating permutations and Euler
numbers. This is joint work with Lilan Dai.